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Asymptotic Behaviour of Continuous Time and State Branching Processes

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  09 April 2009

Zeng-Hu Li
Zeng-Hu Li
Affiliation:
Department of Mathematics Beijing Normal UniversityBeijing 100875 P. R.China e-mail: [email protected]
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Abstract

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We prove some limit theorems for contiunous time and state branching processes. The non-degenerate limit laws are obtained in critical and non-critical cases by conditioning or introducing immigration processes. The limit laws in non-critical cases are characterized in terms of the cononical measure of the cumulant semigroup. The proofs are based on estimates of the cumulant semigroup derived from the forward and backward equations, which are easier than the proffs in the classical setting.

Keywords

branching processescontinuous timecontinuous sateconditioningimmigrationlimit theorem

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 1 , February 2000 , pp. 68 - 84
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Athreya, K. B. and Ney, P. E., Branching processes (Springer, Berlin, 1972).Google Scholar
[2]Bingham, N. H., ‘Continuous branching processes and spectral positivity’, Stochastic Process. Appl. 4 (1976), 217–242.CrossRefGoogle Scholar
[3]Foster, J. H., ‘A limit theorem for a branching process with state-dependent immigration’, Ann. Math. Statist. 42 (1971), 1773–1776.Google Scholar
[4]Grey, D. R., ‘Asymptotic behavior of continuous time, continuous state-space branching processes’, J. Appl. Probab. 11 (1974), 669–677.CrossRefGoogle Scholar
[5]Harris, T. E., ‘Some mathematical models for branching processes’, in: Second Symposium on Probabilily and Statistics (ed. Neyman, J.) (University of California Press, 1951) pp. 305–328.Google Scholar
[6]Kawazu, K. and Watanabe, S., ‘Branching processes with immigration and related limit theorems’, Theory Probab. Appl. 16 (1971), 36–54.Google Scholar
[7]Kesten, H., Ney, P. and Spitzer, F., ‘The Galton-Watson process with mean one and finite variance’, Theory Probab. Appl. 11 (1966), 513–540.CrossRefGoogle Scholar
[8]Lamperti, J. and Ney, P., ‘Conditioned branching processes and their limiting diffusions’, Theory Probab. Appl. 13 (1968), 128–139.Google Scholar
[9]Mitov, K. V., Grishechkin, S. A. and Yanev, N. M., ‘Limit theorems for regenerative processes’, C. R. Acad. Bulgare Sci. 49 (1996), 29–32.Google Scholar
[10]Mitov, K. V. and Yanev, N. M., ‘Bellman-Harris branching processes with state-dependent immigration’, J. Appl. Probab. 22 (1985), 757–765.Google Scholar
[11]Mitov, K. V. and Yanev, N. M., ‘Bellman-Harris branching processes with a special type of state-dependent immigration’, Adv. Appl. Probab. 21 (1989), 270–283.Google Scholar
[12]Pakes, A. G., ‘Revisiting conditional limit theorems for mortal simple branching processes’, Bernoulli (to appear).Google Scholar
[13]Pakes, A. G., ‘A branching process with a state dependent immigration component’, Adv. Appl. Probab. 3 (1971), 301–314.Google Scholar
[14]Pakes, A. G., ‘Some results on non-supercritical Galton-Watson processes with immigration’, Math. Biosci. 24 (1975), 71–92.Google Scholar
[15]Pakes, A. G., ‘On the age distribution of a Markov chain’, J. Appl. Probab. 15 (1978), 65–77.Google Scholar
[16]Pakes, A. G., ‘Some limit theorems for Jirina processes‘, Period. Math. Hung. 10 (1979), 55–66.Google Scholar
[17]Pakes, A. G., ‘Some limit theorems for continuous-state branching processes’, J. Austral. Math. Soc. (Ser. A) 44 (1988), 71–87.Google Scholar
[18]Pakes, A. G. and Trajstman, A. C., ‘Some properties of continuous-state branching processes, with applications to Bartoszynski's virus model, Adv. Appl. Probab. 17 (1985), 23–41.CrossRefGoogle Scholar
[19]Pinsky, M. A., ‘Limit theorems for continuous state branching processes with immigration’, Bull. Amer Math. Soc. 78 (1972), 242–244.CrossRefGoogle Scholar
[20]Seneta, E. and Vere-Jones, D., ‘On the asymptotic behavior of branching processes with continuous state space’, Z Wahrsch. Verw. Gebiete 10 (1968), 212–225.CrossRefGoogle Scholar
[21]Seneta, E. and Vere-Jones, D., ‘On a problem of Jirina concerning continuous state branching processes’, Czechoslovak Math. J. 19 (1969), 277–283.CrossRefGoogle Scholar
[22]Yamazato, M., ‘Some results on continuous time branching processes with state-dependent immigration’, J. Math. Soc. Japan 27 (1975), 479–496.CrossRefGoogle Scholar